With their Obama repeal efforts failed and with customers howling over rising costs, House Republicans went back to the drawing board Wednesday looking to come up with new solutions to boost consumer choices and free businesses from the 2010 law’s punishments.

The Ways and Means Committee debated bills that would let Americans use Obamacare subsidies to buy cheaper plans outside the law’s “exchange” markets, creating an escape hatch for health people who can’t afford robust coverage that is getting more expensive.

Mr. Brady also pushed bills that expand the use of tax-advantaged health savings accounts, giving people more incentive to save for their own care, and another bill to postpone penalties big companies would face if they didn’t provide coverage for employees. That bill would also waive past violations.

Republicans are looking for a counterpunch to liberal efforts to use health care to get supporters to the polls in November.

“Republicans are trying to show action in anticipation of Democratic attacks during the campaign,” said Darrell West, director of governance studies at the Brookings Institution.

Democratic leaders see health care as their No. 1 issue heading into the midterms, pointing to surveys that show Obamacare is more popular than ever and that Republicans will be responsible for any disruptions in the insurance markets, since they control the levers of power in Washington.

Senate Minority Leader Charles E. Schumer said the administration’s decision Tuesday to slash funding for groups who offer in-person sign-up assistance comes on top of moves to cut off critical insurer reimbursements, repeal the mandate that goads healthy people into the marketplace and crack open the door to bare-bones plans that might siphon much-needed customers out of Obamacare’s marketplace.

“Yesterday’s news should remind us President Trump remains ruthlessly committed to tearing down our health care system,” Mr. Schumer said. “He won’t admit it at his rallies. He won’t dare talk about it but that probably is the most important thing he’s doing in terms of effect on the American people and we’re not going to let him hide it.”

House Democrats said the GOP’s latest push will not undo the damage.

For instance, they said expanding health savings accounts will mainly benefit people who have money to spend on health care, yet want to shield the cash from taxation.

“Most of the legislation today does very little for the average American,” said Rep. Richard Neal, Massachusetts Democrat.

Mr. Trump and his GOP allies counter that they’re offering a needed off-ramp for consumers trapped by Obamacare into buying lavish but expensive plans they don’t want.

“Obamacare’s been sabotaged by the Democrats who wrote it in secret without Republicans support or input,” Mr. Brady said, combining the GOP’s Obama-era complaints with the Democrats’ favored barb against Mr. Trump.

Across the Capitol, Sen. Lamar Alexander is leading a series of hearings on the underlying costs of health care, saying that should be the starting point for solutions.

Mr. Trump also released a blueprint for slashing drug costs — a key concern for voters — though several drug companies decided to raise prices, anyway, on July 1.

“They want to make sure they don’t get blamed for voter’s dissatisfaction over rising health costs,” Mr. West said. “The action on opioids and drug pricing also is part of the broader positioning on medical costs.”

Mr. Trump sparred this week with pharmaceutical giant Pfizer, which recently raised prices on 100 drugs, and took a victory lap after the company decided to defer the increases until Jan. 1. Democrats said the relief is temporary, however, so the administration only managed to score “cheap PR points.”